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Math : Trigonometry

 

Contents for Trigonometry Module 2 : Trigonometric Functions and Identities

Values of Trigonometric Functions for Common Angles

It is useful to know the values of the trigonometric functions for certain common angles. The values of the trigonometric functions for angles 0, 30o, 45o, 60o and 90o are given in the table below.

 

Angle A
0
30o
45o
60o
90o
sin A
0
½
1/√2
√3/2
1
cos A
1
√3/2
1/√2
½
0
tan A
0
 1/√3
1
√3
 ∞

 

It is possible to obtain the values in the above table for 0, 45o and 90o by using the definitions from Trigonometry Module 1.

 

The values in the above table can be easily remembered using the following mnemonic. The values of sin 0, sin 30o, sin 45o, sin 60o and sin 90o are simply given by the square roots of 0/4, 1/4, 2/4, 3/4 and 4/4. The values of cos 0, cos 30o, cos 45o, cos 60o and cos 90o are given by the square roots of 4/4, 3/4, 2/4, 1/4 and 0/4. Obviously, the values of tan for any angle are obtained by dividing the sine value by the cosine value.

Important Trigonometric Identities

Some important trigonometric identities relating functions of a single angle (say, A) are given below.

 

sin2 A + cos2 A = 1

... (1)
 1 + tan2 A = sec2 A ... (2)
   1+ cot2 A = cosec2 A  ... (3)

 

 

These identities are useful in simplification and solution of problems. Their proofs are given below.

  • Based on the right-angled triangle in the figure alongside, sin A = a / b and cos A = c / b.
    Therefore sin2 A + cos2 A = (a2 + c2)/b2.
    By Pythagoras Theorem, a2 + c2 = b2 in a right-angled triangle.
    Thus sin2 A + cos2 A = 1 and equation (1) is proved.

     

  • On dividing both sides of equation (1) by cos2 A,
    we obtain tan2 A + 1 = 1 / cos2 A = sec2 A. Equation (2) is proved.

     

  • On dividing both sides of equation (1) by sin2 A,
    we obtain 1 + cot2 A = 1/sin2 A = cosec2 A. Equation (3) is proved.

Cofunction Property

The cofunction property is very important and is given below.

 

sin (90° − A) = cos A ... (4)
cos (90° − A) = sin A ... (5)
tan (90° − A) = cot A ... (6)

 

 

The cofunction property is easy to prove by starting with the fact that the sum of the three angles of a triangle is always 180o, i.e., A + B + C = 180o. For a right-angled triangle, angle B is 90o and A + C = 90o. On using 90o - C = A, the following are readily obtained:

  • sin (90o - A) = sin C = c / b = cos A
  • cos (90o - A) = cos C = a / b = sin A
  • tan (90o - A) = tan C = c / a = cot A

     

Many such properties exist in trigonometry that make problem-solving easier.

Practice Exercise for Trigonometry Module 2: Trigonometric Functions and Identities

 

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